 Solution

An elliptic curve has good reduction of an elliptic curve at $p$ if it admits a Weierstrass equation of an elliptic curve over ${\mathbb{Z}}_p$ whose reduced projective cubic is nonsingular. Equivalently its minimal discriminant is a $p$-adic unit.

For the given integral short equation,It therefore directly supplies good reduction outside $2,3,5$. These three primes cannot be rescued by changing the model. An admissible change of Weierstrass model changes the discriminant by a twelfth power, so its valuation changes by a multiple of twelve. The displayed valuations $4,7,4$ are not congruent to zero modulo twelve. No integral model can have unit discriminant at any of those primes. The good primes are exactly $p\notin \{2,3,5\}$.